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:'''Click [[ECE-QE_AC3-2011_solusion-3|here]] to view student [[ECE-QE_AC3-2011_solusion-3|answers and discussions]]''' | :'''Click [[ECE-QE_AC3-2011_solusion-3|here]] to view student [[ECE-QE_AC3-2011_solusion-3|answers and discussions]]''' | ||
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| − | '''Part 4.''' | + | '''Part 4.''' (20 pts) |
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| + | <font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue} \text{ Consider the following model of a discrete-time system, }</math></span></font> | ||
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| + | <math>x\left ( k+1 \right )=2x\left ( k \right )+u\left ( k \right ), x\left ( 0 \right )=0, 0\leq k\leq 2</math><br> | ||
| + | |||
| + | <math>\color{blue}\text{Use the Lagrange multiplier approach to calculate the optimal control sequence}</math><br> | ||
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| + | <math>\left \{ u\left ( 0 \right ),u\left ( 1 \right ), u\left ( 2 \right ) \right \}</math> | ||
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| + | <math>\color{blue}\text{that transfers the initial state } x\left( 0 \right) \text{ to } x\left( 3 \right)=7 \text{ while minimizing the performance index}</math><br> <math>J=\frac{1}{2}\sum\limits_{k=0}^2 u\left ( k \right )^{2}</math><br> | ||
Revision as of 05:38, 28 June 2012
ECE Ph.D. Qualifying Exam in Automatic Control (AC), Question 3, August 2011
Question
Part 1. 20 pts
$ \color{blue} \text{ Consider the optimization problem, } $
$ \text{maximize} -x_{1}^{2}+x_{1}-x_{2}-x_{1}x_{2} $
$ \text{subject to } x_{1}\geq0, x_{2}\geq0 $
$ \color{blue}\left( \text{i} \right) \text{ Characterize feasible directions at the point } x^{*}=\left[ \begin{array}{c} \frac{1}{2} \\ 0 \end{array} \right] $
$ \color{blue}\left( \text{ii} \right) \text{Write down the second-order necessary condition for } x^{*} \text{. Does the point } x^{*} \text{ satisfy this condition?} $
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Part 2.
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Part 3.
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Part 4. (20 pts)
$ \color{blue} \text{ Consider the following model of a discrete-time system, } $
$ x\left ( k+1 \right )=2x\left ( k \right )+u\left ( k \right ), x\left ( 0 \right )=0, 0\leq k\leq 2 $
$ \color{blue}\text{Use the Lagrange multiplier approach to calculate the optimal control sequence} $
$ \left \{ u\left ( 0 \right ),u\left ( 1 \right ), u\left ( 2 \right ) \right \} $
$ \color{blue}\text{that transfers the initial state } x\left( 0 \right) \text{ to } x\left( 3 \right)=7 \text{ while minimizing the performance index} $
$ J=\frac{1}{2}\sum\limits_{k=0}^2 u\left ( k \right )^{2} $
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Part 5. (20 pts)
$ \color{blue} \text{ Consider the following optimization problem, } $
$ \text{optimize} \left(x_{1}-2\right)^{2}+\left(x_{2}-1\right)^{2} $
$ \text{subject to } x_{2}- x_{1}^{2}\geq0 $
$ 2-x_{1}-x_{2}\geq0 $
$ x_{1}\geq0. $
$ \color{blue} \text{The point } x^{*}=\begin{bmatrix} 0 & 0 \end{bmatrix}^{T} \text{ satisfies the KKT conditions.} $
$ \color{blue}\left( \text{i} \right) \text{Does } x^{*} \text{ satisfy the FONC for minimum or maximum? Where are the KKT multipliers?} $
$ \color{blue}\left( \text{ii} \right) \text{Does } x^{*} \text{ satisfy SOSC? Carefully justify your answer.} $
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